Why do we use +1 and -1 for marginal decision boundaries in SVM

Question

Why do we use +1 and -1 for marginal decision boundaries in SVM

user1825567

2022年5月18日 21:36

While using support vector machines (SVM), we encounter 3 types of lines (for a 2D case). One is the decision boundary and the other 2 are margins:

Why do we use $+1$ and $-1$ as the values after the $=$ sign while writing the equations for the SVM margins? What's so special about $1$ in this case?

For example, if $x$ and $y$ are two features then the decision boundary is: $ax+by+c=0$. Why are the two marginal boundaries represented as $ax+by+c=+1$ and $ax+by+c=-1$?

Topic svm

Category Data Science

Rui Li · Accepted Answer · 2022年5月18日 21:36

This is just mathematical convenience.

Suppose we have $W*X+b \ge k$ for positive points and $W*X+b \le -k$ for negative points. If we scale the equation with $1 \over k$, $W'*X+b' \ge 1$, where $W'= {W \over k}$, $b'= {b \over k}$. This doesn't change the optimization target for $W$ and $b$.

Jan van der Vegt · Accepted Answer · 2017年5月31日 16:25

It's important for the optimization formulation of the SVM that $y_i=\{-1,1\}$ which is why it makes sense to also output $y=\{-1,1\}$. If we look at the soft-margin linear SVM we want to minimize:

$\left[\frac{1}{n}\sum_{i=1}^n\max{(0,1-y_i(w\cdot x_i+b))}\right]+\lambda\| w\| ^2$

The $y_i$ is either +1 or -1 which flips the hyperplane in the soft-margin definition of the problem.

Why do we use +1 and -1 for marginal decision boundaries in SVM

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